where \(\frac{r}{(\deg P)!}\)  is the coefficient of the term \(m^{\mathrm {deg}\,P}\)  in \(P(m)\) .
Notice that \(\mathcal {Z}^\prime \)  is a projective \(S\) -scheme with a \(\mathrm {SL}(V)\) -action and the group \(\mathrm {SL}(V)\)  preserves the map \(\mathcal {Z}^\prime \rightarrow S\)  with the trivial action on \(S\) . Due to [[1]}, Proposition 7] or [[2]}, Lemma 1.13], GIT-(semi)stable points of the total space \(\mathcal {Z}^\prime \)  restricted on any geometric point \(s\)  of \(S\)  are exactly those GIT-(semi)stable points of the fiber of \(\mathcal {Z}^\prime \rightarrow S\)  over \(s\) . Then one can obtain GIT-(semi)stable points of \(\mathcal {Z}^\prime \)  via the restriction on fibers over all geometric points. Therefore, it follows from the similar argument in [[3]}, Section 4.2] that
